Question

The function
$f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x$

• A. increases in $\left[\frac{1}{2}, \infty\right)$
• B. increases in $\left(-\infty, \frac{1}{2}\right]$
• C. decreases in $\left[\frac{1}{2}, \infty\right)$
• D. decreases in $\left(-\infty, \frac{1}{2}\right]$

Answer 1

Answer: (A)

$f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x$
$f'(x)=\left(2 x^{2}-x\right)-2 \cos x+2 \cos x-\sin x(2 x-1)$
$=(2 x-1)(x-\sin x)$
for $x>0, x-\sin x>0$
$x<0, x-\sin x<0$
for $x \in(-\infty, 0] \cup\left[\frac{1}{2}, \infty\right), f'(x) \geq 0$
for $x \in\left[0, \frac{1}{2}\right], f'(x) \leq 0$
$\Rightarrow f(x)$ increases in $\left[\frac{1}{2}, \infty\right)$